Optimal binary linear codes of dimension at most seven

Binary linear codes with at most 4 weights

For the past decades, linear codes with few weights have been widely studied, since they have applications in space communications, data storage and cryptography. In this paper, a class of binary linear codes is constructed and their weight distribution is determined. Results show that they are at most 4-weight linear codes. Additionally, these codes can be used in secret sharing schemes. Index...

Optimal Linear Codes Over GF(7) and GF(11) with Dimension 3

Let $n_q(k,d)$ denote the smallest value of $n$ for which there exists a linear $[n,k,d]$-code over the Galois field $GF(q)$. An $[n,k,d]$-code whose length is equal to $n_q(k,d)$ is called {em optimal}. In this paper we present some matrix generators for the family of optimal $[n,3,d]$ codes over $GF(7)$ and $GF(11)$. Most of our given codes in $GF(7)$ are non-isomorphic with the codes pre...

Binary Optimal Linear Rate 1/2 Codes

Weight hierarchies of binary linear codes of dimension 4

The weight hierarchy of a binary linear [n; k] code C is the sequence (d1; d2; : : : ; dk) where dr is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes. The possible weight hierarchies in each class are given. For one class the details of the proofs are included. c © 2001 Elsevier Science B.V. All rights reserved.

Binary Linear Codes with Optimal Scaling and Quasi-Linear Complexity

We present the first family of binary codes that attains optimal scaling and quasi-linear complexity, at least for the binary erasure channel (BEC). In other words, for any fixed δ > 0, we provide codes that ensure reliable communication at rates within ε > 0 of the Shannon capacity with block length n = O(1/ε2+δ), construction complexity Θ(n), and encoding/decoding complexity Θ(n log n). Furth...